Minimal belief and negation as failure. A feasible approach. Antje Beringer and Torsten Schaub. FG Intellektik, TH Darmstadt, Alexanderstrae 10,

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1 Minimal belief and negation as failure: A feasible approach Antje Beringer and Torsten Schaub FG Intellektik, TH Darmstadt, Alexanderstrae 10, D-6100 Darmstadt, Germany fantje,torsteng@intellektikinformatikth-darmstadtde Abstract Lifschitz introduced a logic of minimal belief and negation as failure, called mbnf, in order to provide a theory of epistemic queries to nonmonotonic databases We present a feasible subsystem of mbnf which can be translated into a logic built on rst order logic and negation as failure, called fonf We give a semantics for fonf along with an extended connection calculus In particular, we demonstrate that the obtained system is still more expressive than other approaches Introduction Lifschitz [1991; 1992] 1 introduced a logic of minimal belief and negation as failure, mbnf, in order to provide a theory of epistemic queries to nonmonotonic databases This approach deals with self-knowledge and ignorance as well as default information From one perspective, mbnf relies on concepts developed by Levesque [1984] and Reiter [1990] for database query evaluation In these approaches, databases are treated as rst order theories, whereas queries may also contain an epistemic modal operator In addition to query-answering from a database, this modal operator allows for dealing with queries about the database From another perspective, Lifschitz' approach relies on the system gk developed by Lin and Shoham [1990], which uses two epistemic operators accounting for the notion of \minimal belief" and \negation as failure" Thus, mbnf can be seen as an extension of gk, which identies their epistemic operator for minimal belief with the ones used by Levesque and Reiter mbnf is very expressive Apart from asking what a database knows, it permits expressing default knowledge and axiomatizing the closed world assumption [Reiter, 1977] and integrity constraints [Kowalski, 1978] Furthermore, Lifschitz [1992] established close relationships to logic programming, default logic [Reiter, 1980] and circumscription [McCarthy, 1980] However, Lifschitz' approach is purely semantical and mainly intended to provide a unifying framework 1 In what follows, we rely on the more recent approach for several nonmonotonic formalisms Consequently, there is no proof theory yet We address this gap by identifying a subsystem of mbnf and translating it into a feasible system by relying on the fact that in many cases negation as failure is expressive enough to account for the dierent modalities in mbnf The resulting system is called fonf (rst order logic with negation as failure) We demonstrate that it provides a versatile approach to epistemic query answering for nonmonotonic databases, which are rst order theories enriched by beliefs and default statements Furthermore, we give a clear semantics of fonf along with an extended connection calculus for fonf Also, we show that fonf is still more expressive than prolog and a competing approach [Reiter, 1990] Minimal belief and negation as failure mbnf deals with an extended rst order language including two independent modal operators B and not B is of epistemic nature and represents the notion of \minimal belief", whereas not captures the notion of \negation as failure" A theory T, or database, is a set of sentences ; denote sentences; F; G denote formulas A positive formula (or theory) does not contain not An objective one contains neither B nor not For instance, given an ornithological database, we can formalize the default that \we believe that birds y, unless there is evidence to the contrary", as 8x(Bbird(x) ^ not:y(x)! By(x)): Now, the idea is to interpret our beliefs by a set of \possible worlds", ie Bbird(Tweety) is true i Tweety is a bird in all possible worlds and not:y(tweety) is true i Tweety ies in some possible world Formally, the truth of a formula is dened wrt a triple (w; W B ; W not ); where w is a rst order interpretation, or simply world, representing \the real world", W B is a set of \possible worlds" dening the meaning of beliefs formalized with B, and W not serves for the same purpose in case of not w; W B and W not share the same universe, but W B and W not do not necessarily include w Intuitively, this means that beliefs need not be consistent with reality Thus, Tweety may be believed to y without actually ying

2 Then, the truth of a formula F in mbnf is dened for the language of mbnf extended by names for all elements of the common universe 2 1 For atomic F; (w; W B ; W not )j= mbnf F i w j= F 2 (w; W B ; W not )j= mbnf :F i (w; W B ; W not ) j= 6 mbnf F 3 (w; W B ; W not )j= mbnf F ^ G i (w; W B ; W not )j= mbnf F and (w; W B ; W not )j= mbnf G 4 (w; W B ; W not )j= mbnf 9XF (X) i for some name, (w; W B ; W not )j= mbnf F () 5 (w; W B ; W not )j= mbnf BF i 8w 0 2 W B : (w 0 ; W B ; W not )j= mbnf F 6 (w; W B ; W not )j= mbnf notf i 9w 0 2 W not : (w 0 ; W B ; W not ) j= 6 mbnf F The denition of a model in mbnf is restricted to the case where W B = W not : Therefore, Lifschitz introduces structures (w; W ), where w is a world and W a set of worlds corresponding to W B and W not In particular, he is only interested in <-maximal structures, where (w; W ) < (w 0 ; W 0 ) i W W 0 ; since they express \the idea of `minimal belief': The larger the set of `possible worlds' is, the fewer propositions are believed" [Lifschitz, 1992] Formally, a model in mbnf is dened by means of a xed-point operator (T; W ); which, given a theory T and a set of worlds W, denotes the set of all <-maximal structures (w; W 0 ) such that T is true in (w; W 0 ; W ) Then, a structure (w; W ) is an mbnf-model of T i (w; W ) 2 (T; W ): In mbnf, theoremhood is only dened for positive formulas: A positive formula F is entailed by T, T j= mbnf F, i F is true in all models of T Thus, query answering is also restricted to positive queries 3 Notice that models of T need not be models of F For instance, Bp is true in all models of B(p ^ q) and, hence, B(p ^ q)j= mbnf Bp: However, no mbnf-model of B(p ^ q) is an mbnf-model of Bp, since none of them is <-maximal in satisfying Bp So, we have to distinguish carefully between formulas in a given theory and formulas being posed as queries to that theory FONF: A feasible approach to MBNF We develop a feasible approach to mbnf by identifying a large subclass of mbnf, which allows for equivalent formalizations in rst order logic plus negation as failure This is the case whenever a theory T is complete for believed sentences, ie whenever we have either T j= mbnf B or T j= mbnf :B for each In this case, rst order logic with negation as failure is strong enough to capture also the notion of \minimal belief" We thus identify a feasible subclass of mbnf for which we provide a translation into fonf, a rst order logic with an additional negation as failure operator not This translation preserves the above notion 2 j= without any subscript denotes rst order entailment 3 As regards mbnf-queries, we rely on this restriction throughout the paper of completeness in the sense, that an mbnf-theory is complete for believed sentences i the corresponding fonf-theory is As mentioned above, we have to distinguish between queries and sentences in a database Accordingly, we dene the following feasible subset of mbnf for queries and databases separately: A feasible query is an mbnf-formula q satisfying: 1 q is positive 2 Each scope of an 9 or : in q is either purely subjective or purely objective 3 If the scope of a : in q is subjective, then it must not contain free variables A feasible database (FDB) is an mbnf-theory containing only rules of the form F 1 ^ : : : ^ F m ^ not F m+1 ^ : : : ^ not F n! F n+1 where for n; m 0 each F i (i = 1; : : : ; n + 1) is { either a disjunction-free mbnf-formula where the scope of : is minimal and objective, 4 { or of the form B(G 1 _ : : : _ G k ) where the G i (i = 1; : : : ; k) are objective formulas F n+1 may also be an unrestricted objective formula These restrictions are not as strong as it seems at rst sight: Even default rules, integrity constraints, and closed world axioms can be formalized within FDBs In [Lifschitz, 1992] an mbnf-formula F is translated into a rst order formula F to relate mbnf- and rst order entailment: A second sort of so-called \world variables" is added to the rst order language; appending one of them to each function and predicate symbol (as an additional argument), and introducing a unary predicate B whose argument is such a world variable A world variables denotes the world in which a certain predicate or function symbol is interpreted and B accounts for the \accessibility" of a world from the actual world However, the translation is insucient for creating a deduction method for mbnf First, it deals only with positive formulas and, therefore, discards a substantial half of mbnf: The modal operator not Second, only rst order entailment carries over to mbnf but not vice versa That is, roughly speaking, even for positive T and, T j= implies T j= mbnf but not vice versa In this sense, the translation is sound but incomplete Our approach addresses this shortcoming by translating feasible queries and databases into fonf This has the following advantages: First, we deal with a much larger subset of mbnf In particular, we can draw nonmonotonic conclusions by expressing B and not by a rst order predicate bel and a negation as failure operator not Second, our translation is truthpreserving That is, for feasible queries and databases, fonf-entailment carries over to mbnf and vice versa 4 Observe that one cannot distribute : over B

3 In this sense, the translation is sound and complete for feasible queries and databases In the sequel, we give this translation and prove that it is truth-preserving Now, fonf-formulas are all formulas that can be built using the connectives and construction rules of rst order logic and the unary operator not The only constraint on fonf-formulas is that variables must not occur free in the scope of not The translation? of feasible mbnf-queries and -databases into fonf-formulas is developed in analogy to [Lifschitz, 1992] We use the predicate bel to translate the mbnf-operator B Then, a feasible mbnf-formula F, ie either a feasible query or a formula belonging to a FDB, is translated into the fonf-formula F? in the following way If F is objective, then 1 F? is obtained by appending the world variable V to each function and predicate symbol in F else (ie if F is non-objective) 2 (:F )? = not F? 3 (F G)? = F? G? for = ^; _ or! 4 (Q F )? = Q F? for Q = 9 or 8 5 (BF )? = 8V (bel(v )! F? ) 6 (not F )? = not(bf )? Observe that feasible queries must not contain not, so that then Condition 6 does not apply The translation? depends on the notion of feasible formulas which obey syntactical restrictions Thus, we have to account for all connectives As an example, translating : _ :B (; objective) into fonf yields :? _ not(8v (bel(v )!? ); which shows that the combination :B is translated using negation as failure, namely not, whereas pure negation : is kept In order to show that this translation is truthpreserving, we look at the semantics of fonf and dene satisability wrt a set of worlds W : W j= fonf i 8w 2 W : (w; W )j= fonf ; where the truth value of a fonf-formula wrt a structure (w; W ) is dened in the following way: If F is objective, then 1 (w; W )j= fonf F i w j= F else (ie if F is non-objective) 2 (w; W )j= fonf :F i (w; W ) j= 6 fonf F 3 (w; W )j= fonf F ^ G i (w; W )j= fonf F and (w; W )j= fonf G 4 (w; W )j= fonf notf i 9w 0 2 W: (w 0 ; W ) j= 6 fonf F fonf can be seen as an extension of extended logic programs [Gelfond and Lifschitz, 1990] Accordingly, fonf-models extend the semantics of extended logic programs to the rst order case: For a fonf-theory T, and a set of worlds W, we develop a set of objective formulas T W from T by 1 deleting all rules, where not occurs in the body while W j= fonf holds 2 deleting all remaining subformulas of the form not Then, W is a fonf-model for T if it consists of all rst order models of T W : A sentence is entailed by a fonf-theory T, T j= fonf, i is true in all fonfmodels of T Then, we obtain the equivalence between query-answering in mbnf and fonf for FDBs Theorem 1 For feasible mbnf-databases T and feasible mbnf-queries : T j= mbnf i T? j= fonf? : As a corollary, we get that the translation? preserves completeness for believed sentences in the above sense The translation proposed in [Lifschitz, 1992] satises only one half of the above result since it only provides completeness for \monotonically answerable queries" Moreover, Lifschitz deals with a more restricted fragment of mbnf, which excludes, for example, the use of negation as failure for database sentences A connection calculus for FONF We develop a calculus for fonf based on the connection method [Bibel, 1987], an armative method for proving the validity of a formula in disjunctive normal form (DNF) These formulas are displayed twodimensionally in the form of matrices A matrix is a set of sets of literals Each element of a matrix represents a clause of a formula's DNF In order to show that a sentence is entailed by a sentence T, we have to check whether :T _ is valid In the connection method this is accomplished by path checking: A path through a matrix is a set of literals, one from each clause A connection is an unordered pair of literals with the same predicate symbol, but dierent signs A connection is complementary if both literals are identical except for their sign Now, a formula, like :T _; is valid i each path through its matrix contains a complementary connection under a global substitution First, we extend the denition of matrices and literals An NM-Literal is an expression not M, where M is an NM-matrix An NM-matrix is a matrix containing normal and NM-literals Although these structures seem to be rather complex, we only deal with normal form matrices, sincenm-literals are treated in a special way during the deduction process The denition of classical normal form matrices relies on the DNF of formulas Here, we deal with formulas in disjunctive fonf-normal form (fonf-dnf) by treating subformulas like not as atoms while transforming formulas into DNF Then these 's are transformed recursively into fonf-dnf and so forth An NM-matrix M F represents a quantier-free fonfformula F in fonf-dnf as follows 1 If F is a literal then M F = fff gg 2 If F = not G then M F = ffnot M G gg 3 If F = F 1 ^ : : : ^ F n then M F = f S n i=1 M F i g

4 4 If F = F 1 _ : : : _ F n then M F = S n i=1 M F i For instance, p(x) _ (q(a) ^ not (r(y ) ^ q(y ) _ p(a))) has the following matrix representation: p(x) not q(a) r(y ) q(y ) p(a) In order to dene a nonmonotonic notion of complementarity, we introduce so-called adjunct matrices They are used for resolving the complex structures in NM-matrices during the deduction process Given an NM-matrix M = fc 1 ; : : : ; C n g where C n = fl 1 ; : : : ; L m ; not Ng the adjunct matrix M N is dened as M N = (M n C n ) [ N, or two-dimensionally: C 1 M : : : C n 1 not L 1 N C 1 M N : : : C n 1 An NM-matrix is NM-complementary if each path through the matrix is NM-complementary A path p is NM-complementary if p contains a connection fk; Lg which is complementary under unication or p contains an NM-literal not N with N being an NM-matrix If the adjunct matrix M N is not NMcomplementary, then p is NM-complementary The deduction algorithm relies on the standard connection method, except that if a path contains an NMliteral then the same deduction algorithm is started recursively with the corresponding adjunct matrix Let M be an NM-matrix and let P M be the set of all paths through M Then the NM-complementarity of M is checked by checking all paths in P M for NMcomplementarity This is accomplished by means of the procedure nmc in the following informal way for a set of paths P 0 and a matrix M 0 nmc(p 0 ; M 0 ) If P 0 = ;, then nmc(p 0 ; M 0 )=\yes" else choose p 2 P 0 { If p is classically complementary with connection fk; Lg and unier, then nmc(p 0 ; M 0 ) = nmc((p 0 fp j fk; Lg 2 pg); M 0 ) { else if there exists an NM-literal not N 2 p such that nmc(p MN ; M N )=\no", then nmc(p 0 ; M 0 )=nmc(p 0 fp j not N 2 pg; M 0 ) else nmc(p 0 ; M 0 )=\no" Initially, nmc is called with P M and M, namely nmc(p M ; M) Then, we obtain the following result Proposition 1 If the algorithm terminates with \yes", then the NM-matrix is NM-complementary N So far, we have only considered quantier-free fonfformulas in fonf-dnf But how can an arbitrary fonf-formula F be treated within this method? First of all, F must be transformed into fonf-skolem normal form, analogously to [Bibel, 1987] We denote the result of fonf-skolemization of a formula F by S(F ) Now, fonf-formulas can be represented as matrices If we have a fonf-database, we require that rules containing free variables in the scope of not (which is actually not allowed in fonf) are replaced by the set of their ground instances before skolemization However, the above algorithm has its limitations due to its simplicity First, in mbnf and fonf, it is necessary to distinguish between sentences occurring in the database and those serving as queries Representing a database together with a query as an NMmatrix removes this distinction Second, the above algorithm cannot deal with fonf-theories possessing multiple fonf-models This requires a separate algorithmic treatment of alternative fonf-models We address this shortcoming by slightly restricting the denition of feasible queries and databases instead of providing a much more complicated algorithm Thus, we introduce determinate queries and databases 5 Determinate queries are feasible mbnfqueries in DNF which are either objective or consist only of one non-objective disjunct Determinate databases are FDBs which do not contain circular sets of rules, like fnot p! Bq; not q! Bpg; because only such circular sets cause multiple models in fonf This restriction is not as serious as it might seem at rst sight: First, non-objective disjunctive queries can be posed by querying the single disjuncts separately Second, we doubt that we loose much expressiveness by forbidding circular rules So, if we consider this kind of databases and queries, we can use the nonmonotonic connection method for query-answering: Proposition 2 For determinate fonf-theories T and fonf-sentences : T j= fonf i the NM-matrix of S(:T _ ) is NM-complementary Together with Theorem 1, we obtain the following Theorem 2 For determinate mbnf-databases T and determinate mbnf-queries : T j= mbnf i the NMmatrix of S(:T? _? ) is NM-complementary Hence, we obtain a deduction method for a quite large subset of mbnf: Given a determinate mbnf-database T and -query, we check whether T j= mbnf holds by 1translating T into T? and into?, 2 replacing free variables with all constants occurring in the database, 3skolemizing :T? _? yielding S(:T? _? ), 4 testing the resulting matrix fornm-complementarity 5 This expression will also be used for the corresponding fonf-queries and -databases

5 Finally, let us consider an example illustrating our approach Consider the following mbnf-database T : B(teaches(anne,bio) _ teaches(sue,bio)) not teaches(x,bio)! : teaches(x,bio) which we write in short notation as B(t(a; b) _ t(s; b)) ^ (not t(x; b)! :t(x; b)): Recall that the second conjunct is considered as an abbreviation for the set of its ground instances Consider the following query: Is it true that Anne doesn't teach biology? This query, say, is formalized in mbnf as :t(a; b) Notice that T and constitute determinate mbnfexpressions Now, we have to verify whether T j= mbnf holds According to the closed world axiom given by not t(x; b)! :t(x; b); saying that a person does not teach biology unless proven otherwise, we expect a positive answer Following the four steps above, we rst translate T and into fonf and obtain the fonf-theory T? 8V bel(v )! t(a; b; V ) _ t(s; b; V ) not [8V bel(v )! t(x; b; V )]! :t(x; b; V ) along with the query? = :t(a; b; V ) Then, the theory is negated yielding :T? After replacing X by the constants a; s and b 6 in :T?, we obtain S(:T? _? ) by fonf-skolemization which is (with Skolem constants w i (i = 1; 2; 3)) bel(v ) ^ :t(a; b; V ) ^ :t(s; b; V ) not [:bel(w 2 ) _ t(a; b; w 2 )] ^ t(a; b; w 1 ) not [:bel(w 3 ) _ t(s; b; w 3 )] ^ t(s; b; w 1 ) :t(a; b; w 1 ): This fonf-formula has the following matrix representation (if we ignore the drawn line) During the proof for the above adjunct matrix a copy of the rst clause has to be generated We get the substitution = fv 1 nw 1 ; V 2 nw 2 g, where V 1 occurs in the rst copy and V 2 in the second one The resulting matrix contains the (non-complementary) path f:t(a; b; w 1 ); :t(s; b; w 2 ); :bel(w 2 ); t(a; b; w 2 ); t(s; b; w 1 ); :t(a; b; w 1 )g: The rst two literals stem from the two copies of the rst clause of the adjunct matrix The four others belong to the remaining clauses of the adjunct matrix Since this path through the adjunct matrix is not complementary, the NM-literal not N 1 in the original matrix is NM-complementary Therefore, all paths through the original matrix are NM-complementary Accordingly, we have proven that T j= mbnf holds and thus that Anne doesn't teach biology In order to illustrate the dierence between queries to and about the database in presence of the closed world assumption (CWA), consider the query, say, Is it known that Anne doesn't teach biology? Now, we expect a negative answer, since the used formalization of the CWA only aects objective formulas It avoids merging propositions about the world (like objective formulas in the database) and propositions about the database, which causes inconsistencies when using the \conventional" CWA [Reiter, 1977] in the presence of incomplete knowledge is formalized in mbnf as B:t(a; b) yielding the skolemized fonf-formula :bel(w 4 )_:t(a; b; w 4 ): Treating and the above formula T according to the four aforementioned steps results in the following matrix with submatrices N 1 and N 2 as dened above: bel(v ) not N 1 not N 2 :bel(w 4 ) :t(a; b; w 4 ) bel(v ) not N 1 not N 2 :t(a; b; w 1 ) :t(a; b; V ) t(a; b; w 1 ) t(s; b; w 1 ) :t(a; b; V ) t(a; b; w 1 ) t(s;b; w 1 ) :t(s; b; V ) :t(s; b; V ) with the submatrices N 1 = [:bel(w 2 ) t(a; b; w 2 )] and N 2 = [:bel(w 3 ) t(s; b; w 3 )] It remains to be checked whether this matrix is NM-complementary in order to prove T j= mbnf : The rst connection starting from the query :t(a; b; w 1 ) is shown by the drawn line It remains to be tested, if all paths through the NM-literal not N 1 are NMcomplementary So, the adjunct matrix has to be built yielding the following matrix (with N 2 as above), which must not be NM-complementary for a successful proof bel(v ) :t(a; b; V ) :t(s; b; V ) :bel(w 2 ) t(a; b; w 2 ) not N 2 t(s; b; w 1 ) :t(a; b; w 1 ) 6 For simplicity, we omit the last case in this example, as it obviously does not contribute to the proof Looking at the three paths without NM-literals, it can be easily seen that at least one of them will never be complementary (regardless of how V is instantiated) Consequently, the matrix is not NM-complementary, which tells us that is not an mbnf-consequence of T That is, even though we were able to derive that Anne does not teach biology, we cannot derive that it is known that Anne does not teach biology This is because the used closed world axioms merely aect what is derivable and not what is known However, observe that the opposite query, namely :B:t(a; b) is answered positively Certainly, we could obtain dierent answers by using dierent closed world axioms The above algorithm has been implemented using a prolog implementation of the connection method The program takes NM-matrices and checks whether they are NM-complementary It consists only of ve prolog-clauses Interestingly, the rst four clauses constitute a full rst order theorem prover, and merely

6 the fth clause deals with negation as failure This extremely easy way of implementation is a benet of the restriction to determinate queries and -databases Conclusion mbnf [Lifschitz, 1992] is very expressive and thus very intractable Therefore, we have presented a feasible approach to minimal belief and negation as failure by relying on the fact that in many cases negation as failure is expressive enough to capture additionally the (nonmonotonic) notion of minimal belief We have identied a substantial subsystem of mbnf: Feasible databases along with feasible queries This subsystem allows for a truth-preserving translation into fonf, a rst order logic with negation as failure However, feasibility has its costs For instance, fonf does not allow for \quantifying-in" not Also, we have given a semantics of fonf by extending the semantics of extended logic programs [Gelfond and Lifschitz, 1990] We have developed an extended connection calculus for fonf, which has been implemented in prolog To our knowledge, this constitutes the rst connection calculus integrating negation as failure We wanted to keep our calculus along with its algorithm as simple as possible, so that it can be easily adopted by existing implementations of the connection method, like setheo [Letz et al, 1992] The preservation of simplicity has resulted in the restriction to determinate theories, which possess only single fonf-models This restriction is comparable with the one found in extended logic programming, where one restricts oneself to well-behaved programs with only one model As a result, we can compute determinate queries to determinate databases in mbnf This subset of mbnf is still expressive enough for many purposes: Apart from asking what a database knows, determinate queries and databases allow for expressing default rules, axiomatizing the closed world assumption and integrity constraints Also, it seems that the restriction to determinate queries and databases can be dropped in the presence of a more sophisticated algorithm treating multiple fonf-models separately Moreover, our approach is still more expressive than others: First, fonf is more expressive than prolog: Since it is build on top of a rst order logic, it allows for integrating disjunctions and existential quantication Second, Reiter [1990] has proposed another approach, in which databases are treated as rst order theories, whereas queries may include an epistemic modal operator As shown in [Lifschitz, 1992], this is equivalent to testing whether BT j= mbnf BF holds for objective theories T and positive formulas F of mbnf Obviously, Reiter's approach is also subsumed by determinate queries and databases, so that we can use our approach to implement his system as well Although we cannot account for mbnf in its entirety, our approach still deals with a very expressive and, hence, substantial subset of mbnf Moreover, from the perspective of conventional theorem proving, our translation has shown how the epistemic facet of negation as failure can be integrated into automated theorem provers To this end, it is obviously possible to implement fonf by means of other deduction methods, like resolution In particular, it remains future work to compare resolution-based approaches to negation as failure to the approach presented here Acknowledgements We thank S Bruning, M Lindner, A Rothschild, S Schaub, and M Thielscher for useful comments on earlier drafts of this paper This work was supported by DFG, MPS (HO 1294/3-1) and by BMfT, TASSO (ITW 8900 C2) References Bibel, W 1987 Automated Theorem Proving, Vieweg, 2nd edition Gelfond, M and Lifschitz, V 1990 Logic programs with classical negation In Proc International Conference on Logic Programming 579{597 Kowalski, R 1978 Logic for data description In Gallaire, H and Minker, J, eds, In Proc Logic and Databases, Plenum 77{103 Letz, R; Bayerl, S; Schumann, J; and Bibel, W 1992 Setheo: A high-performance theorem prover Journal on Automated Reasoning Levesque, H 1984 Foundations of a functional approach to knowledge representation Articial Intelligence 23:155{212 Lifschitz, V 1991 Nonmonotonic databases and epistemic queries In Myopoulos, J and Reiter, R, eds, In Proc International Joint Conference on Articial Intelligence, Morgan Kaufmann 381{386 Lifschitz, V 1992 Minimal belief and negation as failure Submitted Lin, F and Shoham, Y 1990 Epistemic semantics for xed{points nonmonotonic logics In Parikh, Rohit, ed, In Proc Theoretical Aspects of Reasoning about Knowledge 111{120 McCarthy, J 1980 Circumscription a form of nonmonotonic reasoning Articial Intelligence 13(1{ 2):27{39 Reiter, R 1977 On closed world data bases In Gallaire, H and Nicolas, J-M, eds, In Proc Logic and Databases, Plenum 119{140 Reiter, R 1980 A logic for default reasoning Articial Intelligence 13(1{2):81{132 Reiter, R 1990 On asking what a database knows In Lloyd, J W, ed, Computational Logic, Springer 96{113